#347011

order	512
nilpotency class	3
isoclinism	isoclinic class of #347002
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#62881	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2015 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,e^b=eh,f^b=fi,g^b=gi,d^c=dh,e^c=ei,f^c=fi,e^d=ei

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, 254207);