#2309862

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

3084 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)(5,6,8,9)(7,10,11,12)(13,14,17,19)(15,16,20,18)(22,25)(23,27);(6,9)(10,12)(13,15,17,20)(14,18,19,16)(23,27)(26,28);(7,11)(10,12)(14,19)(15,20)(21,22)(23,26)(24,25)(27,28);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,16,17,18)(14,20,19,15)(21,23)(22,26)(24,27)(25,28);(5,7)(6,10)(8,11)(9,12)(13,14)(15,18)(16,20)(17,19)(21,22)(23,26)(24,25)(27,28)

Polycyclic presentation:

2^9|a^2=f,b^2=h,d^2=fghi,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.

Relationships

maximal subgroup of	#72256741, #72256754, #72256776
maximal normal subgroup of	#72256741, #72256754, #72256776

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2217058);