#948425

order	512
nilpotency class	2
isoclinism	isoclinic class of #917169
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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2256 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)(6,10)(7,12)(21,22)(23,30)(24,27)(25,33)(26,29)(28,34)(31,35)(32,36);(6,10)(7,12)(14,17)(18,20)(21,23,26,24)(22,27,29,30)(25,31,34,32)(28,35,33,36);(5,6,9,10)(7,11,12,8)(13,14,16,17)(15,18,19,20)(21,22)(23,27)(24,30)(25,28)(26,29)(31,35)(32,36)(33,34);(1,2,3,4)(5,7)(6,11)(8,10)(9,12)(21,24,26,23)(22,27,29,30)(25,32,34,31)(28,35,33,36);(5,8,9,11)(6,7,10,12)(13,15,16,19)(14,18,17,20)(21,25)(22,28)(23,31)(24,32)(26,34)(27,35)(29,33)(30,36)

Polycyclic presentation:

2^9|b^2=f,c^2=h,d^2=fgi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,d^c=df

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, 855621);