#2103748

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
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×#2456	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2561 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=fgh,c^2=f,d^2=h,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef

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, 2010944);