#7613717

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

2117 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=fg,d^2=h,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,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, 7520913);