#115230

order	512
nilpotency class	3
isoclinism	isoclinic class of #115222
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

847 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,e^2=h,f^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,f^a=fi,c^b=cf,f^b=fh,g^b=gh,d^c=di,e^c=eh,f^c=fh,g^c=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, 22426);