#7517399

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
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

2057 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)(7,12)(8,10)(9,11);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,14)(15,20)(16,25)(17,22)(18,27)(19,23)(21,26)(24,28);(2,4)(13,15)(14,20)(16,19)(17,26)(18,24)(21,22)(23,25)(27,28);(2,4)(5,8,9,7)(6,10,11,12)(13,16,19,15)(14,21,23,28)(17,26,18,24)(20,27,25,22);(6,11)(10,12)(13,17,19,18)(14,22,23,27)(15,24,16,26)(20,28,25,21)

Polycyclic presentation:

2^9|b^2=f,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,h^d=hi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308194, #72308272, #72308463
maximal normal subgroup of	#72308194, #72308272, #72308463

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7424595);