#7611909

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
prime radical	#45241	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2050 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)(7,9)(8,10)(11,12)(13,14,19,21)(15,22,25,17)(16,18,26,20)(23,28,27,24);(1,2,4,3)(5,7)(6,9)(8,11)(10,12)(13,15,19,25)(14,17,21,22)(16,23,26,27)(18,24,20,28);(6,10)(7,11)(13,16)(14,18)(15,23)(17,24)(19,26)(20,21)(22,28)(25,27);(6,10)(9,12)(13,17,20,27,19,22,18,23)(14,15,16,24,21,25,26,28);(1,3,4,2)(13,18,19,20)(14,16,21,26)(15,24,25,28)(17,23,22,27)

Polycyclic presentation:

2^9|a^2=i,b^2=f,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, 7519105);