#6380945

order	512
nilpotency class	3
isoclinism	isoclinic class of #61071
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

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×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1986 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=i,e^2=gi,f^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,c^b=ci,d^b=df,e^b=ei,f^b=fi,f^d=fi

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6288141);