#2329536

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

1836 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)(6,9)(7,11)(13,14,17,20)(15,19,16,18)(22,26)(23,29)(24,31)(34,36);(5,6)(7,10)(8,9)(11,12)(13,15,17,16)(14,18,20,19)(22,26)(23,29)(28,33)(30,35);(5,7,8,11)(6,10,9,12)(13,14)(15,19)(16,18)(17,20)(21,22)(23,27)(24,28)(25,26)(29,32)(30,34)(31,33)(35,36);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(21,23,25,29)(22,27,26,32)(24,30,31,35)(28,34,33,36);(5,7)(6,10)(8,11)(9,12)(13,16,17,15)(14,19,20,18)(21,24,25,31)(22,28,26,33)(23,30,29,35)(27,34,32,36)

Polycyclic presentation:

2^9|a^2=fgh,b^2=fgh,c^2=gh,d^2=ghi,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, 2236732);