#192536

order	512
nilpotency class	3
isoclinism	isoclinic class of #18132
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#385
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

832 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,e^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fh,e^b=eh,f^b=fi,e^c=ei,f^c=fh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 99732);