#117833

order	512
nilpotency class	3
isoclinism	isoclinic class of #117832
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

599 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)(3,7)(4,6)(5,12)(8,11)(9,15)(10,14)(13,16)(17,18)(19,28)(20,32,35,25)(21,36)(22,26)(23,27)(24,30)(29,45,47,42)(31,41)(33,39,46,48)(34,43,38,44)(37,40);(2,6)(3,9,10,13)(7,15,14,16)(8,12)(17,19,21,30,23,31,22,37)(18,24,26,28,27,40,36,41)(20,33,34,29,35,46,38,47)(25,42,43,48,32,45,44,39);(1,3)(2,7)(4,10)(5,13)(6,14)(8,15)(9,11)(12,16)(17,20)(18,25)(19,29)(21,34)(22,38)(23,35)(24,39)(26,43)(27,32)(28,42)(30,46)(31,47)(33,37)(36,44)(40,48)(41,45)

Polycyclic presentation:

2^9|a^2=g,b^2=di,d^2=h,f^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,c^b=cf,f^b=fh,d^c=di,f^c=fhi,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, 25029);