#7613632

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518952
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	#59550	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2013 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)(7,10)(8,9)(11,12)(13,14,20,15)(16,26,27,22)(17,21,28,25)(18,24,19,23);(2,4)(6,9)(10,12)(13,15,20,14)(16,26,27,22)(17,21,28,25)(18,23,19,24);(5,7,8,11)(6,10,9,12)(13,16,18,17,20,27,19,28)(14,21,23,26,15,25,24,22);(1,2,3,4)(13,17,18,27,20,28,19,16)(14,22,23,21,15,26,24,25);(6,9)(10,12)(16,27)(17,28)(21,25)(22,26)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fg,d^2=hi,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,d^c=di,e^c=ei,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7520828);