#182680

order	512
nilpotency class	3
isoclinism	isoclinic class of #17597
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	C₂³×C₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1270 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,d^2=h,e^2=h,b^a=be,c^a=ch,e^a=eh,g^a=ghi,e^b=ei,d^c=dhi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 89876);