#232514

order	512
nilpotency class	3
isoclinism	isoclinic class of #9980
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#392
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#392
prime radical	#3006	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1278 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)(3,6)(4,7)(5,8)(9,13)(10,12)(11,14)(15,16)(17,18,23,28)(19,29,32,40)(20,25,36,43)(21,27,22,26)(24,33,39,31)(30,45,46,41)(34,48,47,38)(35,44,37,42);(6,12)(13,16)(17,19)(18,24)(20,34)(21,33)(22,31)(23,32)(25,41)(26,40)(27,29)(28,39)(30,35)(36,47)(37,46)(38,42)(43,45)(44,48);(1,3)(2,6)(4,9)(5,10)(7,13)(8,12)(11,15)(14,16)(17,20)(18,25)(19,30)(21,35)(22,37)(23,36)(24,38)(26,42)(27,44)(28,43)(29,41)(31,34)(32,46)(33,47)(39,48)(40,45);(1,4,5,11)(2,7,8,14)(3,9,10,15)(6,13,12,16)

Polycyclic presentation:

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

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, 139710);