#7562116

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518936
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×#120 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2312 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fgi,d^2=fg,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cgi,d^b=dh,g^b=gi,h^b=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72263899, #72263991, #72264050
maximal normal subgroup of	#72263899, #72263991, #72264050
maximal quotient of	#72264987

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7469312);