#1398425

order	512
nilpotency class	2
isoclinism	isoclinic class of #1393169
automorphism	?
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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2470 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)(7,9)(8,11)(21,22)(23,24)(25,28)(26,27)(29,32)(30,31)(33,36)(34,35);(5,6,10,12)(7,11,9,8)(14,17)(18,20)(22,27)(23,29)(28,35)(30,36);(5,7,10,9)(6,11,12,8)(13,14,16,17)(15,18,19,20)(21,23,26,29)(22,24,27,32)(25,30,34,36)(28,31,35,33);(1,2)(3,4)(5,8)(6,7)(9,12)(10,11)(21,24)(22,23)(25,33)(26,32)(27,29)(28,36)(30,35)(31,34);(5,9,10,7)(6,8,12,11)(13,15,16,19)(14,18,17,20)(21,25)(22,28)(23,30)(24,31)(26,34)(27,35)(29,36)(32,33)

Polycyclic presentation:

2^9|b^2=g,c^2=gh,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^c=df,e^d=ef

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, 1305621);