#3464140

order	512
nilpotency class	2
isoclinism	isoclinic class of #3332717
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2859 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(2,4)(10,11)(12,16)(17,18,22,25)(19,26,20,23)(21,24,30,29)(27,32,28,31);(6,8)(10,11)(14,15)(17,19,22,20)(18,23,25,26)(21,27,30,28)(24,31,29,32);(5,6)(7,8)(9,10,13,11)(12,14,16,15)(17,20,22,19)(18,23,25,26)(21,27,30,28)(24,32,29,31);(1,2,3,4)(9,11,13,10)(12,15,16,14)(17,18)(19,26)(20,23)(21,29)(22,25)(24,30)(27,31)(28,32);(9,12,13,16)(10,14,11,15)(17,21)(18,24)(19,27)(20,28)(22,30)(23,31)(25,29)(26,32)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=fg,d^2=fi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280828, #72281124, #72281269
maximal normal subgroup of	#72280828, #72281124, #72281269

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3371336);