#6500198

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500186
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3540	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3200 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)(13,14,19,31,18,23,16,21)(15,24,25,20,28,33,27,35)(17,22,32,42,30,37,29,36)(26,38,39,34,40,43,41,44);(5,6)(7,11)(8,10)(9,12)(13,15)(14,20)(16,25)(17,26)(18,28)(19,27)(21,33)(22,34)(23,35)(24,31)(29,39)(30,40)(32,41)(36,43)(37,44)(38,42);(2,4)(5,7)(6,9)(8,12)(10,11)(13,16,18,19)(14,21,23,31)(15,25,28,27)(17,29,30,32)(20,33,35,24)(22,36,37,42)(26,39,40,41)(34,43,44,38);(7,12)(9,11)(13,17)(14,22)(15,26)(16,29)(18,30)(19,32)(20,34)(21,36)(23,37)(24,38)(25,39)(27,41)(28,40)(31,42)(33,43)(35,44);(6,10)(9,11)(17,30)(22,37)(26,40)(29,32)(34,44)(36,42)(38,43)(39,41)

Polycyclic presentation:

2^9|a^2=fi,c^2=i,f^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,f^b=fi,d^c=dh,e^d=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, 6407394);