#1295207

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
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

2298 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)(5,6,9,12)(7,10,11,8)(13,14,16,18)(15,19,20,17)(21,22,26,27)(23,31,32,28)(24,34,35,29)(25,33,36,30);(5,7,9,11)(6,10,12,8)(13,14,16,18)(15,17,20,19)(22,27)(23,32)(29,34)(30,33);(5,6)(7,10)(8,11)(9,12)(13,14)(15,17)(16,18)(19,20)(21,23,26,32)(22,28,27,31)(24,33,35,30)(25,29,36,34);(1,2,3,4)(5,6)(7,10)(8,11)(9,12)(21,24,26,35)(22,29,27,34)(23,33,32,30)(25,28,36,31);(5,8,9,10)(6,11,12,7)(13,15,16,20)(14,17,18,19)(21,25,26,36)(22,30,27,33)(23,29,32,34)(24,31,35,28)

Polycyclic presentation:

2^9|a^2=g,b^2=fg,c^2=f,d^2=ghi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,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, 1202403);