#225529

order	512
nilpotency class	3
isoclinism	isoclinic class of #19883
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	D₄²
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

622 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 132725);