#355035

order	512
nilpotency class	3
isoclinism	isoclinic class of #20671
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#382
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

958 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226664, #72226671, #72226675
maximal normal subgroup of	#72226664, #72226671, #72226675

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 262231);