#6399580

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×#3269	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2190 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)(15,20)(16,29)(17,22)(18,32)(19,24)(21,30)(23,33)(25,39)(26,35)(27,42)(28,37)(31,38)(34,40)(36,43)(41,44);(6,9)(7,11)(13,15,19,28)(14,20,24,37)(16,25,17,26)(18,27,33,43)(21,34,38,44)(22,35,29,39)(23,36,32,42)(30,40,31,41);(13,16,19,17)(14,21,24,38)(15,25,28,26)(18,31,33,30)(20,34,37,44)(22,32,29,23)(27,41,43,40)(35,42,39,36);(5,7,8,11)(6,10,9,12)(13,17,19,16)(14,22,24,29)(15,26,28,25)(18,31,33,30)(20,35,37,39)(21,32,38,23)(27,41,43,40)(34,42,44,36);(1,3,4,2)(6,9)(10,12)(16,17)(21,38)(22,29)(25,26)(30,31)(34,44)(35,39)(40,41)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=i,d^2=fi,e^2=h,g^2=i,b^a=bf,c^a=cg,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, 6306776);