#7515535

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	#2416	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1840 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=f,d^2=i,e^2=i,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=cg,d^b=dh,g^b=gi,h^b=hi,g^c=gi,h^c=hi,e^d=ei,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, 7422731);