#192685

order	512
nilpotency class	3
isoclinism	isoclinic class of #18682
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	C₂×#121	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

970 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)(5,6,10,14)(7,12,16,19)(8,13,9,18)(11,17,20,15)(21,22,26,30)(23,31,32,35)(24,27,25,29)(28,33,36,34);(1,3,4,2)(5,7)(6,11)(8,15)(9,17)(10,16)(12,18)(13,19)(14,20)(22,27)(24,25)(28,35)(29,30)(31,36)(33,34);(5,7)(6,12)(8,17)(9,15)(10,16)(11,18)(13,20)(14,19)(21,23)(22,28)(24,34)(25,33)(26,32)(27,35)(29,31)(30,36);(5,8,10,9)(6,13,14,18)(7,15,16,17)(11,19,20,12)(23,32)(28,36)(31,35)(33,34)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 99881);