#7541934

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518944
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#25421	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1840 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=gi,e^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^b=ei,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7449130);