#2042430

order	512
nilpotency class	2
isoclinism	isoclinic class of #2031997
automorphism	?
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	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3260 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:

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

Polycyclic presentation:

2^9|b^2=fgi,d^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72267558
maximal normal subgroup of	#72267558
maximal quotient of	#72267560

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1949626);