#7604242

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518920
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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1807 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)(13,14,20,22)(15,19,17,18)(16,24,27,23)(21,26,28,25);(1,2,4,3)(5,7)(6,10)(8,11)(9,12)(13,15)(14,19)(16,21)(17,20)(18,22)(23,25)(24,26)(27,28);(5,8,9,6)(7,11,12,10)(13,16,18,25,20,27,19,26)(14,21,17,24,22,28,15,23);(7,12)(10,11)(13,17,18,22,20,15,19,14)(16,24,25,28,27,23,26,21);(1,3,4,2)(13,18,20,19)(14,15,22,17)(16,25,27,26)(21,23,28,24)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7511438);