#7604404

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1835 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7511600);