#947063

order	512
nilpotency class	2
isoclinism	isoclinic class of #917169
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	C₂×#2410	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

4161 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|c^2=fg,d^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,d^c=df

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, 854259);