#372899

order	512
nilpotency class	2
isoclinism	isoclinic class of #14715
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

1136 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)(7,10)(8,12)(9,11)(13,14)(15,17)(16,18)(19,20)(21,22,26,29,25,28,30,23)(24,32,35,31,34,27,36,33);(2,4)(5,7,9,8)(6,10,11,12)(13,15,16,19)(14,17,18,20)(21,23,30,28,25,29,26,22)(24,33,36,27,34,31,35,32);(6,11)(10,12)(14,18)(15,19)(21,24)(22,27)(23,31)(25,34)(26,35)(28,32)(29,33)(30,36);(2,4)(5,8,9,7)(6,12,11,10)(24,34)(27,32)(31,33)(35,36)

Polycyclic presentation:

2^9|a^2=h,b^2=fh,d^2=fg,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^c=di

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 280095);