#1087290

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	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3146 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

(2,4)(5,6)(7,11)(8,12)(9,10)(13,14,17,20)(15,19,16,18)(21,22,24,26)(23,25,28,27);(6,10)(8,11)(21,22)(23,27)(24,26)(25,28);(6,10)(7,12)(13,15,17,16)(14,18,20,19)(21,22,24,26)(23,25,28,27);(1,2)(3,4)(5,7)(6,11)(8,10)(9,12)(13,16,17,15)(14,19,20,18)(21,22)(23,25)(24,26)(27,28);(5,8,9,11)(6,7,10,12)(13,14)(15,18)(16,19)(17,20)(21,23,24,28)(22,25,26,27)

Polycyclic presentation:

2^9|a^2=fg,c^2=fg,d^2=fgh,e^2=f,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, 994486);