#364423

order	512
nilpotency class	3
isoclinism	isoclinic class of #196290
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2488 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)(8,20,24,31)(9,14)(10,25)(11,15)(13,23,32,22)(16,33,34,30)(17,27)(18,28)(19,29)(21,36,35,26);(2,4)(5,7)(6,12)(8,21,24,35)(9,17)(10,18)(11,19)(13,30,32,33)(14,27)(15,29)(16,22,34,23)(20,36,31,26)(25,28);(5,8)(6,13)(7,16)(9,22)(10,23)(11,24)(12,26)(14,31)(15,32)(17,21)(18,35)(19,34)(20,25)(27,33)(28,30)(29,36);(2,4)(5,9,11,10)(6,14,15,25)(7,17,19,18)(8,22,24,23)(12,27,29,28)(13,31,32,20)(16,21,34,35)(26,33,36,30)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312528, #72312585, #72312596
maximal normal subgroup of	#72312528, #72312585, #72312596

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 271619);