#705314

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
automorphism	?
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

2662 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=fghi,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,d^c=df

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 612510);