#7125410

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3318	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1935 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:

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

Polycyclic presentation:

2^9|b^2=fi,c^2=hi,d^2=hi,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=ehi,d^c=dh,e^c=ei,f^d=fi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283384, #72283589
maximal normal subgroup of	#72283384, #72283589

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7032606);