#192698

order	512
nilpotency class	3
isoclinism	isoclinic class of #18712
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)(7,12)(8,14)(9,13)(10,15)(11,16)(17,20)(18,19)(21,22)(23,31)(24,27)(25,30)(26,29)(28,34)(32,35)(33,36);(1,3,4,2)(5,7)(6,11)(8,16)(9,18)(10,17)(12,13)(14,20)(15,19)(22,27)(24,25)(28,35)(29,30)(31,36)(33,34);(5,7)(6,12)(8,18)(9,16)(10,17)(11,13)(14,19)(15,20)(21,23)(22,28)(24,34)(25,33)(26,32)(27,35)(29,36)(30,31);(5,8,10,9)(6,13,15,14)(7,16,17,18)(11,12,19,20)(22,29)(23,32)(27,30)(33,34)

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=hi,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dh,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, 99894);