#181334

order	512
nilpotency class	3
isoclinism	isoclinic class of #181131
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#95
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1171 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=h,e^2=hi,f^2=hi,g^2=hi,b^a=be,e^a=eh,g^a=ghi,c^b=ch,f^b=fi,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88530);