#7613626

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518952
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

2186 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)(7,10)(8,9)(11,12)(13,14,17,19)(15,21,23,28)(16,25,24,20)(18,27,26,22);(2,4)(6,9)(10,12)(14,19)(15,16)(18,26)(20,21)(23,24)(25,28);(5,7,8,11)(6,10,9,12)(13,15,18,24,17,23,26,16)(14,20,22,21,19,25,27,28);(1,2,3,4)(13,16,18,15,17,24,26,23)(14,21,22,25,19,28,27,20);(6,9)(10,12)(14,19)(15,23)(16,24)(22,27)

Polycyclic presentation:

2^9|a^2=i,c^2=fgi,d^2=h,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,d^c=di,e^c=ei,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308261, #72308472, #72308843
maximal normal subgroup of	#72308261, #72308472, #72308843

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7520822);