#470994

order	512
nilpotency class	3
isoclinism	isoclinic class of #461203
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2644
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

645 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)(3,5)(4,6)(7,8)(9,10)(11,12,23,28)(13,29)(14,18)(15,20)(16,22,27,24)(17,32)(19,25)(21,37,38,26)(30,34,33,39)(31,36)(35,40);(1,3,4,7)(2,5,6,8)(9,11,14,22,15,23,25,24)(10,16,19,28,20,27,18,12)(13,21,31,34,32,38,40,39)(17,33,36,37,29,30,35,26);(9,12,25,27,15,28,14,16)(10,11,18,24,20,23,19,22)(13,30,40,37,32,33,31,26)(17,34,35,21,29,39,36,38);(2,6)(5,8)(9,13,14,31,15,32,25,40)(10,17,18,35,20,29,19,36)(11,21,22,34,23,38,24,39)(12,26,27,30,28,37,16,33)

Polycyclic presentation:

2^9|a^2=efg,b^2=efh,c^2=ef,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,g^b=gi,d^c=de,e^c=ei,f^c=fi,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 378190);