#234904

order	512
nilpotency class	3
isoclinism	isoclinic class of #234892
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1182 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,7,12)(3,13,4,15)(5,11,6,10)(8,16,9,14)(17,18)(19,23)(20,24)(21,25)(22,26)(27,35)(28,36)(29,37)(30,39)(31,40)(32,41)(33,42)(34,43)(38,45)(44,47)(46,48);(1,3)(2,8)(4,7)(5,16)(6,14)(9,12)(10,15)(11,13)(17,19,22,29)(18,23,26,37)(20,27,31,45)(21,32,34,28)(24,38,40,35)(25,41,43,36)(30,46,33,44)(39,47,42,48);(1,4,7,3)(2,9,12,8)(5,16,6,14)(10,15,11,13)(17,20)(18,24)(19,27)(21,33)(22,31)(23,35)(25,42)(26,40)(28,46)(29,45)(30,34)(32,44)(36,48)(37,38)(39,43)(41,47);(17,21,22,34)(18,25,26,43)(19,28,29,32)(20,30,31,33)(23,36,37,41)(24,39,40,42)(27,44,45,46)(35,47,38,48)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,d^2=i,e^2=h,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,g^a=gi,c^b=cg,d^b=di,e^b=eh,d^c=di,e^c=ei,f^c=fh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 142100);