#6399798

order	512
nilpotency class	3
isoclinism	isoclinic class of #61643
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	C₂×#547	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2162 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)(13,14,19,25)(15,27,30,21)(16,23,17,32)(18,26,20,24)(22,34,39,31)(28,37,33,40)(29,38,35,41)(36,44,42,43);(6,9)(7,11)(13,15)(14,21)(16,28)(17,33)(18,35)(19,30)(20,29)(22,36)(23,40)(24,41)(25,27)(26,38)(31,44)(32,37)(34,43)(39,42);(13,16,19,17)(14,22,25,39)(15,28,30,33)(18,31,20,34)(21,36,27,42)(23,26,32,24)(29,43,35,44)(37,41,40,38);(5,7,8,11)(6,10,9,12)(13,17,19,16)(14,23,25,32)(15,28,30,33)(18,31,20,34)(21,37,27,40)(22,26,39,24)(29,44,35,43)(36,41,42,38);(1,3,4,2)(6,9)(10,12)(13,18,19,20)(14,24,25,26)(15,29,30,35)(16,31,17,34)(21,38,27,41)(22,23,39,32)(28,43,33,44)(36,37,42,40)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6306994);