#401082

order	512
nilpotency class	3
isoclinism	isoclinic class of #395238
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23402	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1711 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)(7,18)(8,13)(9,24)(10,16)(11,17)(12,20)(14,26)(15,22)(19,29)(21,28)(23,32)(25,30)(27,34)(31,35)(33,36);(2,4)(5,7,10,20,11,8,22,21)(6,12,15,18,17,28,16,13)(9,19,25,33,26,23,35,34)(14,27,30,32,24,36,31,29);(2,4)(5,8)(6,13)(7,11)(9,23)(10,21)(12,15)(14,29)(16,28)(17,18)(19,26)(20,22)(24,32)(25,34)(27,30)(31,36)(33,35);(5,9,10,25,11,26,22,35)(6,14,15,30,17,24,16,31)(7,19,20,33,8,23,21,34)(12,27,18,32,28,36,13,29)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286994, #72287345, #72287360
maximal normal subgroup of	#72286994, #72287345, #72287360

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 308278);