#62624

order	256
nilpotency class	2
isoclinism	isoclinic class of H(5,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₈ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₁₆	C₂⁴
commutator, socle	C₂	C₂⁴×C₈
Frattini, cosocle	C₈	C₂⁵
prime radical	#585	C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆ ◁ G
LECS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

1019 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 64 points:

(1,2,5,10,6,11,20,30,7,12,21,31,22,32,42,52)(3,8,15,25,16,26,38,48,14,24,36,46,37,47,56,61)(4,9,18,28,19,29,41,51,17,27,39,49,40,50,57,62)(13,23,34,44,35,45,55,60,33,43,53,58,54,59,63,64);(1,3)(2,8)(4,13)(5,15)(6,16)(7,14)(9,23)(10,25)(11,26)(12,24)(17,33)(18,34)(19,35)(20,38)(21,36)(22,37)(27,43)(28,44)(29,45)(30,48)(31,46)(32,47)(39,53)(40,54)(41,55)(42,56)(49,58)(50,59)(51,60)(52,61)(57,63)(62,64);(1,4)(2,9)(3,13)(5,18)(6,19)(7,17)(8,23)(10,28)(11,29)(12,27)(14,33)(15,34)(16,35)(20,41)(21,39)(22,40)(24,43)(25,44)(26,45)(30,51)(31,49)(32,50)(36,53)(37,54)(38,55)(42,57)(46,58)(47,59)(48,60)(52,62)(56,63)(61,64);(4,17)(9,27)(13,33)(18,39)(19,40)(23,43)(28,49)(29,50)(34,53)(35,54)(41,57)(44,58)(45,59)(51,62)(55,63)(60,64);(3,14)(8,24)(13,33)(15,36)(16,37)(23,43)(25,46)(26,47)(34,53)(35,54)(38,56)(44,58)(45,59)(48,61)(55,63)(60,64)

Polycyclic presentation:

2^8|a^2=f,f^2=g,g^2=h,e^b=eh,d^c=dh

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#11710557, #11710558, #62624⋊C₅, …
maximal subgroup of	#152006, #152007, #495677, …
maximal normal subgroup of	#152006, #152007, #495677, …
maximal quotient of	#7622419, #7622430, #7622520, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 55612);