#364622

order	512
nilpotency class	3
isoclinism	isoclinic class of #23892
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3010	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2385 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:

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

Polycyclic presentation:

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

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, 271818);