#850707

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2270 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:

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

Polycyclic presentation:

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

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, 757903);