#7564950

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518952
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	#59038	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2014 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72263890, #72263899, #72263940
maximal normal subgroup of	#72263890, #72263899, #72263940
maximal quotient of	#72264281, #72264289

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7472146);