#192216

order	512
nilpotency class	3
isoclinism	isoclinic class of #18446
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	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#385
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

969 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

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, 99412);