#385535

order	512
nilpotency class	3
isoclinism	isoclinic class of #30441
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1157 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,c^2=ef,d^2=ef,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,g^b=gi,e^c=ei,f^c=fi,g^d=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 292731);