#7611921

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

2141 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)(13,14,18,19)(15,20,22,17)(16,24,27,21)(23,26,25,28);(1,2,4,3)(5,7)(6,9)(8,11)(10,12)(13,15,18,22)(14,20,19,17)(16,25,27,23)(21,26,24,28);(6,10)(7,11)(13,16,18,27)(14,21,19,24)(15,23,22,25)(17,26,20,28);(6,10)(9,12)(13,17,19,15,18,20,14,22)(16,26,24,23,27,28,21,25);(1,3,4,2)(16,27)(21,24)(23,25)(26,28)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=i,d^2=h,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

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, 7519117);