#2503790

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	C₂×#2678	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3182 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=hi,e^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	#72332008, #72332021, #72332049
maximal normal subgroup of	#72332008, #72332021, #72332049
maximal quotient of	#72333061

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2410986);