#6370674

order	512
nilpotency class	2
isoclinism	isoclinic class of #3532
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator	C₂²	C₂³×C₄²
socle	C₂³	C₂⁴×C₄
prime radical	C₂⁴	C₂³×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2026 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=f,e^2=fg,g^2=i,b^a=bf,c^a=ci,d^a=dfi,c^b=cf,e^c=ei

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6277870);