#6370615

order	512
nilpotency class	2
isoclinism	isoclinic class of #3532
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator	C₂²	C₂³×C₄²
socle	C₂³	C₂⁴×C₄
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2074 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6277811);