#451233

order	512
nilpotency class	3
isoclinism	isoclinic class of #417541
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2598
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

808 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:

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

Polycyclic presentation:

2^9|a^2=efgh,b^2=ef,c^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,h^b=hi,d^c=di

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 358429);