#401035

order	512
nilpotency class	3
isoclinism	isoclinic class of #393989
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2397
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1732 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)(7,15,18,27)(8,12)(9,13)(10,14)(11,17,23,21)(16,29,30,36)(19,25)(20,26)(22,28)(24,31,34,33)(32,35);(5,7,9,21,10,18,22,17)(6,11,13,27,14,23,28,15)(8,16,20,33,19,30,32,31)(12,24,26,36,25,34,35,29);(8,19)(11,23)(12,25)(15,27)(16,30)(20,32)(26,35)(31,33);(2,4)(5,8,9,20,10,19,22,32)(6,12,13,26,14,25,28,35)(7,16,21,33,18,30,17,31)(11,24,27,36,23,34,15,29)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72256282, #72256284, #72256302
maximal normal subgroup of	#72256282, #72256284, #72256302

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 308231);