#2542895

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×#2410	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3972 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)(5,6)(7,9)(8,10)(11,12)(13,14,17,19)(15,20,16,18)(22,24)(26,28);(1,2,4,3)(5,6,8,10)(7,9,11,12)(13,15,17,16)(14,18,19,20)(23,27)(26,28);(2,3)(7,11)(9,12)(15,16)(18,20)(21,22,25,24)(23,26,27,28);(5,6)(7,9)(8,10)(11,12)(13,14)(15,20)(16,18)(17,19)(21,23)(22,26)(24,28)(25,27);(1,3,4,2)(5,7,8,11)(6,9,10,12)(13,16,17,15)(14,18,19,20)(21,24,25,22)(23,28,27,26)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=ghi,e^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	#72263377, #72263386, #72263396
maximal normal subgroup of	#72263377, #72263386, #72263396

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2450091);