#7551176

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
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	#45018	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2142 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)(13,14,19,24)(15,25,27,28)(16,26,20,21)(17,23,18,22);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15,18,26,19,27,17,21)(14,20,23,25,24,16,22,28);(6,9)(10,11)(13,16)(14,21)(15,23)(17,25)(18,28)(19,20)(22,27)(24,26);(2,4)(5,7,8,12)(6,10,9,11)(13,17,19,18)(14,22,24,23)(15,26,27,21)(16,28,20,25);(13,15,17,21,19,27,18,26)(14,20,22,28,24,16,23,25)

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=g,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^d=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, 7458372);