#1197563

order	512
nilpotency class	2
isoclinism	isoclinic class of #1082705
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

2044 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)(6,8)(7,11)(13,14,17,16)(15,18,19,20)(21,22)(23,30)(24,27)(25,33)(26,29)(28,34)(31,35)(32,36);(15,19)(18,20)(21,23,26,24)(22,27,29,30)(25,31,34,32)(28,35,33,36);(5,6,9,8)(7,10,11,12)(13,15)(14,18)(16,20)(17,19)(21,24,26,23)(22,27,29,30)(25,31,34,32)(28,36,33,35);(1,2,3,4)(5,7)(6,10)(8,12)(9,11)(21,25)(22,28)(23,31)(24,32)(26,34)(27,35)(29,33)(30,36);(5,8,9,6)(7,12,11,10)(13,16,17,14)(15,20,19,18)(21,22)(23,30)(24,27)(25,28)(26,29)(31,36)(32,35)(33,34)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1104759);