#6399767

order	512
nilpotency class	3
isoclinism	isoclinic class of #61609
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2156 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=f,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,g^a=gi,d^b=df,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6306963);