#928541

order	512
nilpotency class	2
isoclinism	isoclinic class of #917169
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	C₂⁴×D₄	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3908 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=ghi,d^2=g,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, 835737);