#346500

order	512
nilpotency class	3
isoclinism	isoclinic class of #346482
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#62869	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2096 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:

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

Polycyclic presentation:

2^9|a^2=h,d^2=hi,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fh,e^b=eh,f^b=fi,g^b=gh,e^c=ei,f^c=fh,e^d=eh

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 253696);