#396270

order	512
nilpotency class	3
isoclinism	isoclinic class of #394127
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1462 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)(7,16)(8,21,22,24)(12,25)(13,27,29,28)(17,30)(18,33,31,34)(19,20)(26,32,35,36);(1,2,4,3)(5,6)(7,17)(8,13)(9,14)(10,23)(11,15)(12,19)(16,30)(18,32)(20,25)(21,27)(22,29)(24,28)(26,34)(31,36)(33,35);(5,7,9,16)(6,12,14,25)(8,18,22,31)(10,20,11,19)(13,26,29,35)(15,30,23,17)(21,33,24,34)(27,32,28,36);(1,3,4,2)(5,8)(6,13)(7,18)(9,22)(10,21)(11,24)(12,26)(14,29)(15,28)(16,31)(17,32)(19,34)(20,33)(23,27)(25,35)(30,36)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286662, #72286995, #72287003
maximal normal subgroup of	#72286662, #72286995, #72287003

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 303466);