#6396109

order	512
nilpotency class	3
isoclinism	isoclinic class of #62283
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	C₂⁵	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1982 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)(16,28)(17,22)(18,32)(19,23)(21,29)(24,38)(25,34)(26,35)(27,42)(30,36)(31,37)(33,39)(40,43)(41,44);(1,2,4,3)(5,7)(6,9)(8,12)(10,11)(13,15)(14,20)(16,24)(17,25)(18,27)(19,26)(21,33)(22,34)(23,35)(28,38)(29,39)(30,40)(31,41)(32,42)(36,43)(37,44);(1,3,4,2)(13,16,18,29,19,30,17,31)(14,21,22,28,23,37,32,36)(15,24,27,39,26,40,25,41)(20,33,34,38,35,44,42,43);(7,12)(9,11)(13,17,19,18)(14,22,23,32)(15,25,26,27)(16,29,30,31)(20,34,35,42)(21,36,37,28)(24,39,40,41)(33,43,44,38);(6,10)(9,11)(15,26)(20,35)(24,40)(25,27)(33,44)(34,42)(38,43)(39,41)

Polycyclic presentation:

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

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, 6303305);