#4416995

order	512
nilpotency class	2
isoclinism	isoclinic class of #4414082
automorphism	Aut(G) order 8388608
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

2098 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)(7,11)(10,12)(15,19)(17,20)(21,22,26,27)(23,25,29,34)(24,31,33,28)(30,36,35,32);(6,9)(10,12)(13,14,16,18)(15,17,19,20)(22,27)(23,29)(28,31)(30,35);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(21,23)(22,25)(24,30)(26,29)(27,34)(28,32)(31,36)(33,35);(13,15,16,19)(14,17,18,20)(21,24,26,33)(22,28,27,31)(23,30,29,35)(25,32,34,36);(5,7)(6,10)(8,11)(9,12)(21,25)(22,29)(23,27)(24,32)(26,34)(28,35)(30,31)(33,36)

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=g,d^2=i,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,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, 4324191);