#6432848

order	512
nilpotency class	3
isoclinism	isoclinic class of #6432843
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×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×#3449	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4338 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:

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

Polycyclic presentation:

2^9|a^2=i,d^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,c^b=ch,f^b=fi,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6340044);