#5022277

order	512
nilpotency class	2
isoclinism	isoclinic class of #5020720
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

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

2167 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=i,d^2=fg,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^c=dg,e^c=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72304039, #72304049, #72304136
maximal normal subgroup of	#72304039, #72304049, #72304136

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4929473);