#2100320

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

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

1804 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fh,c^2=h,d^2=g,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2007516);