#376195

order	512
nilpotency class	3
isoclinism	isoclinic class of #219491
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2328
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23422	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1618 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226670, #72226673, #72226674, …
maximal normal subgroup of	#72226670, #72226673, #72226674, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 283391);