#1434835

order	512
nilpotency class	2
isoclinism	isoclinic class of #1418353
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

2420 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=gi,c^2=f,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,e^d=ef

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1342031);