#115331

order	512
nilpotency class	3
isoclinism	isoclinic class of #115318
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	C₂×#121	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

840 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2,4,8,5,7,13,15)(3,9,11,16,10,6,12,14)(17,18,22,26,23,28,21,27)(19,29,31,24,32,39,37,40)(20,25,35,42,36,43,34,44)(30,41,45,38,46,48,33,47);(6,9)(14,16)(17,19,23,32)(18,24,28,40)(20,33,36,45)(21,37,22,31)(25,41,43,48)(26,39,27,29)(30,34,46,35)(38,44,47,42);(1,3)(2,6)(4,12)(5,10)(7,9)(8,16)(11,13)(14,15)(17,20)(18,25)(19,30)(21,34)(22,35)(23,36)(24,38)(26,42)(27,44)(28,43)(29,41)(31,45)(32,46)(33,37)(39,48)(40,47)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=h,f^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,c^b=cf,d^b=dh,f^b=fh,d^c=di,f^c=fh,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 22527);