#873657

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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2099 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=ghi,d^2=fg,e^2=fgh,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, 780853);