#7563530

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
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1691 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

(5,6,9,12)(7,11,8,10)(13,14)(15,21)(16,30)(17,35)(18,24)(19,25)(20,26)(22,32)(23,29)(27,38)(28,37)(31,36)(33,41)(34,40)(39,42)(43,44);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,15)(14,21)(16,31)(17,19)(18,28)(20,29)(22,38)(23,26)(24,37)(25,35)(27,32)(30,36)(33,43)(34,42)(39,40)(41,44);(13,16,19,33,20,34,18,32)(14,22,24,40,26,41,25,30)(15,27,28,42,29,43,17,31)(21,36,35,44,23,39,37,38);(2,4)(13,17,18,29,20,28,19,15)(14,23,25,35,26,21,24,37)(16,31,32,43,34,42,33,27)(22,39,30,44,41,36,40,38);(5,8,9,7)(6,11,12,10)(13,18,20,19)(14,24,26,25)(15,28,29,17)(16,32,34,33)(21,37,23,35)(22,40,41,30)(27,42,43,31)(36,38,39,44)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=gi,d^2=g,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,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, 7470726);