#127048

order	512
nilpotency class	3
isoclinism	isoclinic class of #1504
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₂×#336
prime radical	C₂⁴	#97
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

410 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)(3,9,10,6)(4,8)(5,7)(11,15,16,13)(12,14)(17,18,22,28,23,29,36,43,24,30,20,26,21,27,37,44)(19,31,34,47,33,42,48,40,35,45,32,46,38,25,39,41);(1,3,4,11,5,10,12,16)(2,6,8,13,7,9,14,15)(17,19)(18,25)(20,34)(21,38)(22,32)(23,33)(24,35)(26,41)(27,45)(28,40)(29,31)(30,42)(36,39)(37,48)(43,46)(44,47);(17,20,23,37,24,22,21,36)(18,26,29,44,30,28,27,43)(19,32,33,39,35,34,38,48)(25,40,31,46,42,41,45,47)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=h,d^2=i,e^2=h,f^2=g,h^2=i,b^a=bd,d^a=dg,c^b=ci,d^b=di,e^b=egi

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 34244);