#182813

order	512
nilpotency class	3
isoclinism	isoclinic class of #17603
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1064 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)(7,15)(8,12)(9,13)(10,14)(11,16)(17,20)(18,19)(21,22,26,30)(23,28,32,35)(24,29,25,34)(27,33,36,31);(5,7,9,17,10,18,8,16)(6,11,12,19,14,20,13,15)(21,23)(22,27)(24,31)(25,33)(26,32)(28,34)(29,35)(30,36);(1,3,4,2)(5,6)(7,11)(8,13)(9,12)(10,14)(15,16)(17,19)(18,20)(21,23)(22,28)(24,33)(25,31)(26,32)(27,34)(29,36)(30,35);(5,8,10,9)(6,12,14,13)(7,16,18,17)(11,19,20,15)(21,24,26,25)(22,29,30,34)(23,31,32,33)(27,35,36,28)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 90009);