#346472

order	512
nilpotency class	3
isoclinism	isoclinic class of #346446
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#62833	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2227 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=h,e^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,e^b=eh,f^b=fi,g^b=gh,e^c=ei,e^d=eh

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72271333, #72271341, #72271343
maximal normal subgroup of	#72271333, #72271341, #72271343

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 253668);