#395431

order	512
nilpotency class	3
isoclinism	isoclinic class of #394034
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3586	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3383 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 302627);