#6394601

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	C₂×#525	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2062 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=h,e^2=i,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

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, 6301797);