#395124

order	512
nilpotency class	3
isoclinism	isoclinic class of #393975
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1868 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312492, #72312596
maximal normal subgroup of	#72312492, #72312596

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 302320);