#7612128

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518920
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1471 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2,5,8)(3,7,4,6)(9,10)(11,13)(12,14)(15,16)(17,18,22,23)(19,21,26,31)(20,27,30,24)(25,32,28,29);(1,3,5,4)(2,6,8,7)(9,11)(10,13)(12,15)(14,16)(17,19,22,26)(18,21,23,31)(20,28,30,25)(24,32,27,29);(10,14)(11,15)(17,20)(18,24)(19,25)(21,29)(22,30)(23,27)(26,28)(31,32);(10,14)(13,16)(17,21,23,26,22,31,18,19)(20,29,27,28,30,32,24,25);(1,4,5,3)(2,7,8,6)(17,18,22,23)(19,21,26,31)(20,27,30,24)(25,32,28,29)

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,d^2=h,e^2=fi,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,e^d=ei

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, 7519324);