#4308613

order	512
nilpotency class	2
isoclinism	isoclinic class of #4307937
automorphism	Aut(G) order 8388608
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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

2378 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)(5,6,8,10)(7,11,12,9)(15,20)(16,18)(24,31)(28,33)(29,35)(34,36);(5,6)(7,9)(8,10)(11,12)(13,14,17,19)(15,18,20,16)(22,26)(27,32)(28,33)(34,36);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,15)(14,18)(16,19)(17,20)(21,22)(23,27)(24,28)(25,26)(29,34)(30,32)(31,33)(35,36);(5,7,8,12)(6,9,10,11)(13,16,17,18)(14,15,19,20)(21,23,25,30)(22,27,26,32)(24,29,31,35)(28,34,33,36);(21,24,25,31)(22,28,26,33)(23,29,30,35)(27,34,32,36)

Polycyclic presentation:

2^9|a^2=f,b^2=fi,d^2=hi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4215809);