#6687078

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667460
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#127 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2110 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=i,c^2=fgi,d^2=f,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,h^c=hi,h^e=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284638, #72308918
maximal normal subgroup of	#72284638, #72308918

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6594274);