#7551117

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518964
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
prime radical	C₂×#3274	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2257 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)(15,24)(16,21)(17,20)(18,27)(19,23)(22,26)(25,28);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15,16,25,19,26,18,17)(14,20,27,22,23,28,21,24);(6,9)(10,11)(13,14)(15,20)(16,27)(17,24)(18,21)(19,23)(22,25)(26,28);(2,4)(5,7,8,12)(6,10,9,11)(13,16,19,18)(14,21,23,27)(15,25,26,17)(20,24,28,22);(13,17,16,15,19,25,18,26)(14,22,27,28,23,24,21,20)

Polycyclic presentation:

2^9|b^2=fi,d^2=gi,e^2=fi,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,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7458313);