#2310552

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

2230 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)(6,7)(8,12)(13,14)(15,17)(16,20)(18,19)(21,22,26,27)(23,28,31,35)(24,25,30,29)(32,34,36,33);(5,6,9,7)(8,10,12,11)(13,15)(14,16)(17,19)(18,20)(22,27)(23,31)(24,30)(33,34);(5,6,9,7)(8,11,12,10)(13,16,18,17)(14,15,19,20)(21,23)(22,28)(24,32)(25,34)(26,31)(27,35)(29,33)(30,36);(1,2,3,4)(5,7,9,6)(8,11,12,10)(13,17,18,16)(14,15,19,20)(21,24,26,30)(22,29,27,25)(23,32,31,36)(28,33,35,34);(5,8)(6,10)(7,11)(9,12)(13,15)(14,16)(17,19)(18,20)(21,25,26,29)(22,30,27,24)(23,33,31,34)(28,32,35,36)

Polycyclic presentation:

2^9|a^2=fg,b^2=fh,c^2=g,d^2=fghi,e^2=fg,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, 2217748);