#7611548

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518944
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
prime radical	#44927	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2196 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)(13,14,17,18)(15,19,23,20)(16,21,26,25)(22,24,28,27);(1,2,4,3)(5,7)(6,9)(8,11)(10,12)(13,15,17,23)(14,19,18,20)(16,24,26,27)(21,28,25,22);(6,10)(7,11)(13,15)(14,20)(16,22)(17,23)(18,19)(21,27)(24,25)(26,28);(6,10)(9,12)(13,16,14,21,17,26,18,25)(15,22,20,27,23,28,19,24);(1,3,4,2)(13,14,17,18)(15,19,23,20)(16,25,26,21)(22,27,28,24)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,d^2=hi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7518744);