#369397

order	512
nilpotency class	3
isoclinism	isoclinic class of #24468
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1444 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,c^2=efg,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,g^b=gi,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 276593);