#3464132

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	C₂×#2539	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3252 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=fg,d^2=i,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	#72280871, #72281124, #72281269
maximal normal subgroup of	#72280871, #72281124, #72281269

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3371328);