#1085144

order	512
nilpotency class	2
isoclinism	isoclinic class of #1082705
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

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

3318 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)(6,8)(13,14,18,21)(15,22,19,16)(17,25,27,20)(23,24,26,28);(10,12)(13,15,18,19)(14,16,21,22)(17,26,27,23)(20,28,25,24);(1,2,4,3)(9,10,11,12)(13,15)(14,16)(17,23)(18,19)(20,24)(21,22)(25,28)(26,27);(5,6)(7,8)(13,16)(14,19)(15,21)(17,24)(18,22)(20,26)(23,25)(27,28);(1,3,4,2)(13,17)(14,20)(15,23)(16,24)(18,27)(19,26)(21,25)(22,28)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 992340);