#7125698

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

2056 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)(22,27)(24,25)(28,35)(29,30)(31,36)(33,34);(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,22,24,29,26,30,25,27)(23,31,34,35,32,28,33,36);(2,4)(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18)(21,23)(22,28)(24,34)(25,33)(26,32)(27,35)(29,36)(30,31);(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19)(22,27)(23,32)(24,25)(28,36)(29,30)(31,35);(6,12)(10,13)(11,20)(18,19)(21,24,26,25)(22,29,30,27)(23,33,32,34)(28,35,31,36)

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=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	#72307379, #72307669, #72307905
maximal normal subgroup of	#72307379, #72307669, #72307905

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7032894);