#7185322

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996719
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 ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1928 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=g,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7092518);