#7533856

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 ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2200 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307366, #72307674, #72308350
maximal normal subgroup of	#72307366, #72307674, #72308350

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7441052);