#4948638

order	512
nilpotency class	2
isoclinism	isoclinic class of #4932388
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂³×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	#14715	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3590 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,d^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72326734, #72326749, #72326760
maximal normal subgroup of	#72326734, #72326749, #72326760

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4855834);