#7604324

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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1920 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)(15,23)(16,19)(17,20)(18,26)(21,24)(22,27)(25,28);(2,3)(5,7)(6,10)(8,11)(9,12)(13,15,18,17,19,24,27,28)(14,20,22,23,16,25,26,21);(1,2,4,3)(5,8,9,6)(7,11,12,10)(13,16,19,14)(15,20,24,25)(17,23,28,21)(18,26,27,22);(1,3,4,2)(7,12)(10,11)(14,16)(20,25)(21,23)(22,26);(13,17,27,15,19,28,18,24)(14,21,26,25,16,23,22,20)

Polycyclic presentation:

2^9|a^2=ghi,b^2=f,c^2=hi,d^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,d^c=di

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, 7511520);