#323131

order	512
nilpotency class	3
isoclinism	isoclinic class of #289819
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

720 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=eghi,c^2=hi,d^2=ei,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dhi,f^c=fi,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 230327);