#7520768

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 ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2141 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257307, #72257569, #72257741, …
maximal normal subgroup of	#72257307, #72257569, #72257741, …
maximal quotient of	#72265029

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7427964);