#7198988

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996375
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	#2535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1829 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)(13,14,18,22,19,23,31,39)(15,24,26,34,27,38,17,20)(16,21,30,37,29,36,42,43)(25,40,28,35,41,44,32,33);(5,6)(7,10)(8,11)(9,12)(13,15,19,27)(14,20,23,34)(16,25,29,41)(17,31,26,18)(21,33,36,35)(22,38,39,24)(28,30,32,42)(37,44,43,40);(1,2,3,4)(5,7,10,6)(8,11,9,12)(13,16,19,29)(14,21,23,36)(15,25,27,41)(17,32,26,28)(18,30,31,42)(20,33,34,35)(22,37,39,43)(24,40,38,44);(5,8,10,9)(6,11,7,12)(13,17,18,15,19,26,31,27)(14,20,22,24,23,34,39,38)(16,28,30,41,29,32,42,25)(21,35,37,44,36,33,43,40);(5,9,10,8)(6,11,7,12)(16,29)(21,36)(25,41)(28,32)(30,42)(33,35)(37,43)(40,44)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gi,d^2=fh,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=dhi,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, 7106184);