#6811674

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#546	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2075 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:

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

Polycyclic presentation:

2^9|b^2=fi,c^2=h,d^2=fi,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6718870);