#2329218

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, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1892 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=h,c^2=fgh,d^2=hi,e^2=h,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, 2236414);