#2308968

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
automorphism	Aut(G) order 1048576
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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1954 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)(5,6,8,10)(7,11,12,9)(14,17)(15,19)(21,22,26,24)(23,27,25,28);(5,6,8,10)(7,9,12,11)(13,14)(15,20)(16,17)(18,19)(21,23,26,25)(22,27,24,28);(5,6)(7,9)(8,10)(11,12)(13,15)(14,18)(16,19)(17,20)(21,24,26,22)(23,27,25,28);(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(21,25,26,23)(22,27,24,28);(5,7)(6,9)(8,12)(10,11)(13,14)(15,20)(16,17)(18,19)

Polycyclic presentation:

2^9|a^2=fh,b^2=fh,c^2=gh,d^2=fgi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef

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, 2216164);