#498926

order	512
nilpotency class	4
isoclinism	isoclinic class of #498890
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	C₂×#3018	C₂
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

2157 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2)(3,4)(6,11)(9,21)(12,24)(13,25)(14,22)(16,23)(19,33)(26,35)(27,31)(28,36)(29,34)(30,32);(5,6)(7,12)(8,13)(9,22)(10,14)(11,21)(15,26)(16,31)(17,27)(18,28)(19,34)(20,29)(23,24)(25,33)(30,36)(32,35);(2,4)(5,7,10,17)(6,12,14,27)(8,18,20,15)(9,23,21,16)(11,24,22,31)(13,28,29,26)(19,30,33,32)(25,36,34,35);(5,8,10,20)(6,13,14,29)(7,15,17,18)(9,19,21,33)(11,25,22,34)(12,26,27,28)(16,30,23,32)(24,35,31,36)

Polycyclic presentation:

2^9|c^2=i,d^2=i,e^2=i,b^a=be,c^a=cf,e^a=ei,g^a=ghi,c^b=cg,e^b=ei,h^b=hi,d^c=di,e^c=eh,g^e=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72256645, #72256646, #72256648
maximal normal subgroup of	#72256645, #72256646, #72256648

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406122);