#33974

order	256
nilpotency class	2
isoclinism	isoclinic class of H(5,𝔽₂)
automorphism	Aut(G) = #72342856 (order 6144) · Inn(G) = C₂⁴ · Out(G) = C₂²×C₄×S₄
homology	Abelianization (H₁) = C₂³×C₁₆ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₁₆	C₂⁴
commutator, socle	C₂	C₂³×C₁₆
prime radical	C₂×#41	C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆ ◁ G
LECS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

231 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#33974⋊C₃
maximal subgroup of	#151904, #152301, #152413, …
maximal normal subgroup of	#151904, #152301, #152413, …
maximal quotient of	#512543, #512545, #512552, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 26962);