#846576

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
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

2758 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)(6,9)(7,11)(13,14,18,20)(15,19,17,16)(21,22)(23,27)(24,28)(25,26);(5,6,10,9)(7,12,11,8)(13,15,18,17)(14,16,20,19)(21,23,26,24)(22,27,25,28);(5,7)(6,8)(9,12)(10,11)(13,16,18,19)(14,15,20,17)(21,24,26,23)(22,27,25,28);(1,2)(3,4)(5,8,10,12)(6,11,9,7)(13,14)(15,16)(17,19)(18,20)(21,25,26,22)(23,27,24,28);(5,9,10,6)(7,12,11,8)(13,17,18,15)(14,19,20,16)

Polycyclic presentation:

2^9|a^2=fh,b^2=fgh,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, 753772);