#6994015

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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 ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2409 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|c^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,f^b=fi,d^c=dh,e^c=ei,f^d=fi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307319, #72307796, #72307866
maximal normal subgroup of	#72307319, #72307796, #72307866

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6901211);