#215008

order	512
nilpotency class	3
isoclinism	isoclinic class of #214768
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#40506	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1059 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:

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

Polycyclic presentation:

2^9|a^2=g,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=ce,d^b=di,f^b=fh,g^b=gh,f^c=fi,g^d=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 122204);