#706323

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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

2686 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)(9,12)(15,19)(18,20)(21,22,26,25)(23,28,31,27)(24,32,34,29)(30,35,33,36);(14,17)(18,20)(21,23,26,31)(22,27,25,28)(24,30,34,33)(29,35,32,36);(1,2)(3,4)(5,6,8,10)(7,9,11,12)(13,14,16,17)(15,18,19,20)(21,23)(22,28)(24,33)(25,27)(26,31)(29,35)(30,34)(32,36);(5,7)(6,9)(8,11)(10,12)(13,15,16,19)(14,18,17,20)(21,24,26,34)(22,29,25,32)(23,30,31,33)(27,35,28,36);(21,25,26,22)(23,27,31,28)(24,32,34,29)(30,35,33,36)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=fi,d^2=h,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, 613519);