#7562416

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514000
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	#36465	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#127 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2251 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gi,d^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,e^b=ei,h^b=hi,h^d=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7469612);