#500420

order	512
nilpotency class	4
isoclinism	isoclinic class of #499795
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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	#2431	C₂²
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

921 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,8)(4,6)(5,7)(9,10,13,15)(11,19,21,32)(12,17,23,27)(14,18,25,28)(16,29,26,22)(20,33,34,40)(24,31,36,38)(30,39,37,35);(1,3,5,4)(2,6,7,8)(10,15)(14,25)(16,26)(17,27)(19,32)(24,36)(30,37)(33,40);(1,4,5,3)(2,6,7,8)(9,11,13,21)(10,16,15,26)(12,20,23,34)(14,22,25,29)(17,30,27,37)(18,19,28,32)(24,35,36,39)(31,33,38,40);(2,7)(6,8)(9,12,13,23)(10,17,15,27)(11,20,21,34)(14,24,25,36)(16,30,26,37)(18,31,28,38)(19,33,32,40)(22,35,29,39)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317526, #72317552, #72317581, …
maximal normal subgroup of	#72317526, #72317552, #72317581, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407616);