#192836

order	512
nilpotency class	3
isoclinism	isoclinic class of #18810
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

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

960 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2,4,3)(5,6)(7,12)(8,13)(9,11)(10,14)(15,18)(16,20)(17,19)(21,22)(23,29)(24,32)(25,27)(26,31)(28,33)(30,36)(34,35);(1,3,4,2)(6,11)(8,9)(12,18)(13,14)(15,17)(19,20)(24,33)(28,32)(29,36)(31,34);(5,7,10,16)(6,12,14,20)(8,17,9,15)(11,18,13,19)(21,23)(22,26)(24,34)(25,30)(27,35)(28,36)(29,32)(31,33);(5,8,10,9)(6,11,14,13)(7,15,16,17)(12,19,20,18)(22,27)(26,35)(28,32)(29,36)

Polycyclic presentation:

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

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, 100032);