#2310623

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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

3237 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)(7,9)(8,10)(11,12)(13,14,16,19)(15,18,20,17)(21,22,24,25)(23,27,28,26);(5,6,8,10)(7,9,11,12)(13,15)(14,17)(16,20)(18,19)(21,22)(23,26)(24,25)(27,28);(7,11)(9,12)(15,20)(17,18)(22,25)(26,27);(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,14)(15,18)(16,19)(17,20)(21,23)(22,26)(24,28)(25,27);(5,7)(6,9)(8,11)(10,12)(13,15)(14,18)(16,20)(17,19)(21,22)(23,26)(24,25)(27,28)

Polycyclic presentation:

2^9|a^2=h,b^2=fh,d^2=hi,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	#72303407
maximal normal subgroup of	#72303407

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2217819);