#470282

order	512
nilpotency class	3
isoclinism	isoclinic class of #469776
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2631
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1175 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)(8,13)(9,11)(10,15)(14,23)(16,27)(17,26)(18,28)(19,34)(20,25)(21,31)(22,24)(29,35)(30,32)(33,36);(6,11,15,14)(12,26,28,24)(13,25,31,30)(27,34,36,29);(5,7,10,18)(6,12,15,28)(8,19,21,35)(9,22,23,17)(11,24,14,26)(13,29,31,34)(16,32,33,20)(25,36,30,27);(2,4)(5,8,9,20,10,21,23,32)(6,13,11,25,15,31,14,30)(7,16,17,19,18,33,22,35)(12,27,26,34,28,36,24,29)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72279407
maximal normal subgroup of	#72279407

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 377478);