#7521394

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	#25148	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1976 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)(7,11)(8,12)(9,10)(13,14,19,22)(15,17,21,18)(16,23,26,28)(20,27,24,25);(1,2)(3,4)(6,10)(11,12)(13,15)(14,17)(16,24)(18,22)(19,21)(20,26)(23,25)(27,28);(14,15)(17,18)(20,23)(21,22)(24,28)(25,27);(2,4)(5,7,9,8)(6,11,10,12)(13,16,19,26)(14,20,22,24)(15,23,21,28)(17,25,18,27);(5,8,9,7)(6,11,10,12)(13,17,19,18)(14,21,22,15)(16,25,26,27)(20,28,24,23)

Polycyclic presentation:

2^9|a^2=i,d^2=fi,e^2=fi,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=cgi,d^b=dh,g^b=gi,h^b=hi,e^c=ei,g^c=gi,h^c=hi,g^d=gi,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, 7428590);