#7125700

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×D₄²	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1921 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)(21,22,26,29)(23,30,32,35)(24,28,25,34)(27,33,36,31);(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,23,25,31,26,32,24,33)(22,27,28,30,29,36,34,35);(2,4)(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18)(21,22)(23,27)(24,34)(25,28)(26,29)(30,31)(32,36)(33,35);(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19)(23,31)(24,25)(27,30)(28,34)(32,33)(35,36);(6,12)(10,13)(11,20)(18,19)(21,24,26,25)(22,28,29,34)(23,31,32,33)(27,35,36,30)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307538, #72307792, #72308233
maximal normal subgroup of	#72307538, #72307792, #72308233

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7032896);