#115079

order	512
nilpotency class	3
isoclinism	isoclinic class of #115074
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	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#60942	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

826 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=g,d^2=i,e^2=hi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fh,c^b=cf,d^b=di,f^b=fi,g^b=ghi,d^c=dh,e^c=ehi,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 22275);