#2318256

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
automorphism	Aut(G) order 1048576
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

1792 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)(5,6,8,9)(7,11,12,10)(21,22)(23,28)(24,32)(25,35)(26,36)(27,29)(30,31)(33,34);(5,6)(7,10)(8,9)(11,12)(14,17)(18,20)(21,23,27,24)(22,28,29,32)(25,33,36,30)(26,31,35,34);(6,9)(10,11)(13,14,16,17)(15,18,19,20)(21,24,27,23)(22,28,29,32)(25,33,36,30)(26,34,35,31);(1,2,3,4)(5,7)(6,10)(8,12)(9,11)(21,25)(22,26)(23,30)(24,33)(27,36)(28,34)(29,35)(31,32);(5,6)(7,10)(8,9)(11,12)(13,15,16,19)(14,18,17,20)(21,26)(22,25)(23,31)(24,34)(27,35)(28,33)(29,36)(30,32)

Polycyclic presentation:

2^9|a^2=f,b^2=fg,c^2=gh,d^2=gi,e^2=fh,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, 2225452);