#365746

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

1330 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)(17,31)(18,22)(19,23)(21,30)(24,38)(25,36)(26,37)(28,43)(29,40)(32,35)(33,39)(34,42)(41,44);(1,3,4,2)(5,7,9,8)(6,10,12,11)(13,15,19,26)(14,20,23,37)(16,28,29,42)(17,25,18,32)(21,38,39,44)(22,35,31,36)(24,33,41,30)(27,43,40,34);(5,7,9,8)(6,11,12,10)(13,16,17,30,19,29,18,33)(14,21,22,27,23,39,31,40)(15,24,25,42,26,41,32,28)(20,34,35,38,37,43,36,44);(5,8,9,7)(6,11,12,10)(16,29)(21,39)(24,41)(27,40)(28,42)(30,33)(34,43)(38,44)

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,c^2=ef,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,g^c=gi

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, 272942);