#7611610

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

2092 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)(7,9)(8,10)(11,12)(13,14)(15,23)(16,27)(17,26)(18,25)(19,21)(20,22)(24,28);(2,4)(5,7)(6,9)(8,11)(10,12)(13,15,23,14,19,25,18,21)(16,17,26,27,22,24,28,20);(6,10)(7,11)(13,16)(14,20)(15,24)(17,25)(18,28)(19,22)(21,27)(23,26);(1,2,3,4)(6,10)(9,12)(14,21)(15,25)(16,22)(26,28);(13,17,18,20,19,24,23,27)(14,22,15,26,21,16,25,28)

Polycyclic presentation:

2^9|b^2=fi,d^2=hi,e^2=f,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, 7518806);