#193110

order	512
nilpotency class	3
isoclinism	isoclinic class of #18992
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	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#385
prime radical	C₂×#121	Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

824 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:

(1,2,5,8)(3,9,10,16)(4,12,13,7)(6,14,15,11)(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);(4,13)(7,12)(9,16)(11,14)(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)(2,6)(4,14)(5,10)(7,16)(8,15)(9,12)(11,13)(17,20)(18,25)(19,29)(21,34)(22,35)(23,36)(24,38)(26,41)(27,42)(28,43)(30,45)(31,47)(32,46)(33,37)(39,44)(40,48);(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=gi,e^2=h,f^2=hi,g^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fhi,e^b=eh,f^b=fi,e^c=ei,f^c=fhi

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, 100306);