#359474

order	512
nilpotency class	3
isoclinism	isoclinic class of #196122
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹

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

3792 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)(7,11)(8,18,21,12)(9,14)(10,13)(15,33,34,25)(16,27)(17,26)(19,36,35,28)(20,32,23,29)(22,31)(24,30);(2,4)(5,7,10,17)(6,11,13,26)(8,19,21,35)(9,22,24,16)(12,28,18,36)(14,31,30,27)(15,23,34,20)(25,32,33,29);(5,8)(6,12)(7,15)(9,23)(10,21)(11,25)(13,18)(14,32)(16,35)(17,34)(19,22)(20,24)(26,33)(27,36)(28,31)(29,30);(2,4)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72302353, #72302623, #72302631
maximal normal subgroup of	#72302353, #72302623, #72302631
maximal quotient of	#72303225

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 266670);