#7533878

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
prime radical	#44932	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2186 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=g,d^2=i,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=dh,h^b=hi,e^c=ei,h^d=hi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7441074);