#2125163

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	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2385 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=fg,c^2=fg,d^2=fg,e^2=fh,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, 2032359);