#7145692

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996671
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 ◁ C₄×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2455 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)(7,11)(10,12)(13,14,18,19,17,22,15,21)(16,24,25,28,26,20,23,27);(6,9)(7,11)(14,19)(15,18)(16,23)(21,22)(25,26)(27,28);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15,17,18)(14,19,22,21)(16,25,26,23)(20,28,24,27);(6,9)(10,12)(13,16,18,25,17,26,15,23)(14,20,19,27,22,24,21,28);(5,7,8,11)(6,10,9,12)(13,15,17,18)(14,19,22,21)(16,23,26,25)(20,27,24,28)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72316934, #72316936, #72316950
maximal normal subgroup of	#72316934, #72316936, #72316950

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7052888);