#7551046

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518944
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

2169 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)(8,9)(10,12)(13,14,20,16)(15,23,22,28)(17,21,19,26)(18,27,24,25);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15,19,25,20,22,17,27)(14,18,21,23,16,24,26,28);(6,9)(10,11)(13,16,20,14)(15,24,22,18)(17,21,19,26)(23,25,28,27);(2,4)(5,7,8,12)(6,10,9,11)(13,17,20,19)(14,21,16,26)(15,25,22,27)(18,28,24,23);(13,18,19,23,20,24,17,28)(14,22,21,27,16,15,26,25)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257034, #72257092, #72257105
maximal normal subgroup of	#72257034, #72257092, #72257105
maximal quotient of	#72307776

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7458242);