#396725

order	512
nilpotency class	3
isoclinism	isoclinic class of #396724
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#397
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1174 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)(5,6,10,14)(7,15,19,31)(8,12,22,29)(9,23,16,13)(11,24,26,18)(17,30,20,34)(21,36,32,28)(25,35,27,33);(5,7,16,24,10,19,9,18)(6,11,23,31,14,26,13,15)(8,20,21,35,22,17,32,33)(12,27,28,34,29,25,36,30);(5,8,9,21,10,22,16,32)(6,12,13,28,14,29,23,36)(7,17,18,33,19,20,24,35)(11,25,15,30,26,27,31,34);(2,4)

Polycyclic presentation:

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

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, 303921);