#7198948

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996355
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1796 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gi,d^2=fhi,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=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, 7106144);