#4032994

order	512
nilpotency class	2
isoclinism	isoclinic class of #3858029
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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

2772 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)(3,6)(4,5)(7,8)(9,10,13,16)(11,15,12,14)(17,18,21,20)(19,23,24,22)(25,26)(27,31)(28,29)(30,32);(1,2,4,5)(3,6,8,7)(11,12)(14,15)(17,19,21,24)(18,22,20,23)(26,29)(30,31);(2,5)(6,7)(9,11,13,12)(10,14,16,15)(17,18,21,20)(19,23,24,22)(25,26,28,29)(27,30,32,31);(1,3)(2,6)(4,8)(5,7)(9,12,13,11)(10,15,16,14)(17,20,21,18)(19,22,24,23)(25,27,28,32)(26,30,29,31);(1,2)(3,7)(4,5)(6,8)(9,10)(11,14)(12,15)(13,16)(17,19)(18,22)(20,23)(21,24)(25,26)(27,31)(28,29)(30,32)

Polycyclic presentation:

2^9|a^2=fg,b^2=hi,c^2=ghi,d^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,e^b=ef,e^c=ef,e^d=ef

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72328208, #72328215, #72328251
maximal normal subgroup of	#72328208, #72328215, #72328251

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3940190);