#400796

order	512
nilpotency class	3
isoclinism	isoclinic class of #393989
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2397
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1330 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)(5,6,9,14,10,12,16,23)(7,15,18,25,19,28,22,11)(8,13,21,27,20,26,30,34)(17,29,32,35,31,36,33,24);(5,7,16,18,10,19,9,22)(6,11,23,15,12,25,14,28)(8,17,30,32,20,31,21,33)(13,24,34,29,26,35,27,36);(6,12)(8,20)(14,23)(17,31)(21,30)(24,35)(29,36)(32,33);(2,4)(5,8,16,30,10,20,9,21)(6,13,23,34,12,26,14,27)(7,17,18,32,19,31,22,33)(11,24,15,29,25,35,28,36)

Polycyclic presentation:

2^9|a^2=e,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 307992);