#402461

order	512
nilpotency class	3
isoclinism	isoclinic class of #394465
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#400
prime radical	C₂⁵	C₂×Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1120 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)(5,6,10,15,9,16,11,14)(7,12,25,27,18,28,19,32)(8,20,23,30,24,31,22,13)(17,29,21,26,33,36,34,35);(5,7,11,25,9,18,10,19)(6,12,14,27,16,28,15,32)(8,21,22,33,24,34,23,17)(13,29,31,26,30,36,20,35);(5,8,11,22,9,24,10,23)(6,13,14,31,16,30,15,20)(7,17,25,21,18,33,19,34)(12,26,27,36,28,35,32,29);(2,4)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257091, #72257140, #72257150
maximal normal subgroup of	#72257091, #72257140, #72257150

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 309657);