#6380522

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 ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2162 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)(8,12)(9,11)(13,14)(15,25)(16,22)(17,24)(18,23)(19,21)(20,28)(26,41)(27,34)(29,36)(30,44)(31,40)(32,37)(33,39)(35,43)(38,42);(5,7,9,8)(6,10,11,12)(13,15,19,29)(14,20,21,34)(16,30,32,43)(17,28,18,27)(22,38,37,41)(23,25,24,36)(26,33,42,31)(35,40,44,39);(5,8,9,7)(6,10,11,12)(14,21)(20,34)(22,37)(23,24)(25,36)(35,44)(38,41)(39,40);(13,16)(14,22)(15,26)(17,31)(18,33)(19,32)(20,35)(21,37)(23,39)(24,40)(25,41)(27,30)(28,43)(29,42)(34,44)(36,38);(13,17,19,18)(14,23,21,24)(15,27,29,28)(16,31,32,33)(20,36,34,25)(22,39,37,40)(26,30,42,43)(35,38,44,41)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gi,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,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, 6287718);