#3823200

order	512
nilpotency class	2
isoclinism	isoclinic class of #3332717
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	#15258	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3055 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=fg,c^2=ghi,d^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,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72336581, #72336586, #72336633
maximal normal subgroup of	#72336581, #72336586, #72336633

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3730396);