#424317

order	512
nilpotency class	3
isoclinism	isoclinic class of #417570
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2408
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1597 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72323003
maximal normal subgroup of	#72323003

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 331513);