#6403716

order	512
nilpotency class	3
isoclinism	isoclinic class of #62307
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

1820 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)(7,11)(8,10)(9,12)(13,14)(15,19)(16,27)(17,21)(18,23)(20,32)(22,29)(24,40)(25,34)(26,35)(28,44)(30,38)(31,39)(33,43)(36,41)(37,42);(5,7)(6,9)(8,12)(10,11)(13,15)(14,19)(16,28)(17,25)(18,26)(20,37)(21,34)(22,36)(23,35)(24,31)(27,44)(29,41)(30,43)(32,42)(33,38)(39,40);(6,10)(9,11)(13,16,29,30,18,31,17,32)(14,20,21,39,23,38,22,27)(15,24,41,42,26,28,25,43)(19,33,34,44,35,37,36,40);(7,12)(9,11)(13,17,18,29)(14,21,23,22)(15,25,26,41)(16,30,31,32)(19,34,35,36)(20,27,38,39)(24,42,28,43)(33,40,37,44);(1,3,4,2)(6,10)(9,11)(15,26)(16,31)(19,35)(20,38)(25,41)(27,39)(30,32)(34,36)

Polycyclic presentation:

2^9|a^2=h,c^2=fgi,d^2=i,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ci,d^b=df,e^b=ei,d^c=di,e^c=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, 6310912);