#7597002

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 ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2212 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)(7,11)(8,9)(10,12)(13,14,19,24)(15,16,21,22)(17,20,27,25)(18,26,28,23);(2,3)(5,6)(7,10)(8,9)(11,12)(13,15,18,17,19,21,28,27)(14,20,23,16,24,25,26,22);(1,2,4,3)(6,9)(10,11)(13,16)(14,21)(15,24)(17,26)(18,25)(19,22)(20,28)(23,27);(1,3,4,2)(5,7,8,12)(6,10,9,11)(15,21)(16,22)(17,27)(20,25);(13,17,28,15,19,27,18,21)(14,22,26,25,24,16,23,20)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=hi,d^2=g,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,e^c=ei,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, 7504198);