#7514437

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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

2895 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)(8,10)(9,12)(13,14)(15,16)(17,21)(18,24)(19,23)(20,25)(22,28)(26,27);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(13,15)(14,20)(16,24)(17,22)(18,25)(19,26)(21,27)(23,28);(14,21)(15,22)(17,19)(23,24)(25,26)(27,28);(2,4)(13,16)(14,22)(15,21)(17,27)(18,20)(19,28)(23,25)(24,26);(6,10)(9,11)(13,17,18,19)(14,23,24,21)(15,22,25,26)(16,27,20,28)

Polycyclic presentation:

2^9|b^2=fi,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=cg,d^b=dh,g^b=gi,h^b=hi,e^c=ei,g^c=gi,h^c=hi,g^d=gi,h^d=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257502, #72257516, #72264241
maximal normal subgroup of	#72257502, #72257516, #72264241

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7421633);