#359782

order	512
nilpotency class	3
isoclinism	isoclinic class of #23990
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2010 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)(7,12)(8,18)(9,11)(10,22)(13,19)(14,21)(15,25)(16,26)(17,24)(20,28)(23,27)(29,35)(30,31)(32,33)(34,36);(6,11)(7,15,17,23)(12,25,24,27)(13,18)(14,22)(16,29,30,34)(26,35,31,36)(28,32);(5,7)(6,12)(8,16)(9,17)(10,23)(11,24)(13,26)(14,25)(15,21)(18,31)(19,30)(20,34)(22,27)(28,35)(29,33)(32,36);(5,8)(6,13)(7,16)(9,19)(10,20)(11,18)(12,26)(14,28)(15,29)(17,30)(21,33)(22,32)(23,34)(24,31)(25,35)(27,36)

Polycyclic presentation:

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

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, 266978);