#7516775

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 ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2308 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)(13,14,18,15)(16,24,27,25)(17,22,19,20)(21,28,23,26);(1,2)(3,4)(6,9)(10,12)(13,15)(14,18)(16,25)(17,20)(19,22)(21,28)(23,26)(24,27);(14,20)(15,22)(17,19)(21,24)(23,25)(26,28);(2,4)(5,7,8,11)(6,10,9,12)(13,16)(14,21)(15,23)(17,26)(18,27)(19,28)(20,24)(22,25);(6,9)(10,12)(13,17,18,19)(14,20,15,22)(16,26,27,28)(21,24,23,25)

Polycyclic presentation:

2^9|a^2=i,d^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,e^b=ei,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	#72257339, #72257569, #72257767, …
maximal normal subgroup of	#72257339, #72257569, #72257767, …
maximal quotient of	#72264775

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7423971);