#7565373

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	#2535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1896 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)(8,12)(9,10)(13,14)(15,19)(16,20)(17,21)(18,22)(23,27)(24,26)(25,28);(2,3)(6,10)(11,12)(13,15,18,16)(14,20,22,19)(17,21,25,28)(23,27,24,26);(1,2,4,3)(5,7,9,8)(6,11,10,12)(13,16,18,15)(14,20,22,19)(17,24,25,23)(21,26,28,27);(1,3,4,2)(5,7,9,8)(6,11,10,12)(13,17)(14,21)(15,23)(16,24)(18,25)(19,27)(20,26)(22,28);(5,8,9,7)(6,11,10,12)(14,22)(15,16)(21,28)(23,24)

Polycyclic presentation:

2^9|b^2=i,c^2=fg,d^2=fgi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,e^b=ei,h^b=hi,e^c=ei,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, 7472569);