#7611524

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 ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2152 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,d^2=hi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,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, 7518720);