#7527349

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514196
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	#36218	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2089 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)(5,6,8,10)(7,9,11,12)(16,24)(20,27)(23,26)(25,28);(1,2,4,3)(5,7)(6,9)(8,11)(10,12)(13,14)(15,21)(16,25)(17,19)(18,22)(20,23)(24,28)(26,27);(6,10)(7,11)(13,15,18,17)(14,19,22,21)(16,23,24,26)(20,25,27,28);(6,10)(9,12)(13,16,18,24)(14,20,22,27)(15,23,17,26)(19,25,21,28);(1,3,4,2)(13,15,18,17)(14,21,22,19)(16,23,24,26)(20,28,27,25)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=i,d^2=i,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cgi,d^b=dh,h^b=hi,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, 7434545);