#7533876

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

2236 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)(13,14,17,21)(15,23,26,19)(16,18,22,20)(24,27,28,25);(1,2,4,3)(7,12)(9,11)(13,15)(14,19)(16,24)(17,26)(18,25)(20,27)(21,23)(22,28);(5,7,8,12)(6,9,10,11)(13,16,17,22)(14,20,21,18)(15,24,26,28)(19,27,23,25);(6,10)(9,11)(14,21)(15,25)(18,20)(19,24)(23,28)(26,27);(1,3,4,2)(14,21)(16,22)(19,23)(24,28)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=g,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, 7441072);