#7564623

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514196
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#124 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2123 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)(5,6,9,10)(7,11,8,12)(13,14,16,19)(15,21,23,18)(17,20,25,28)(22,27,24,26);(2,3)(6,10)(11,12)(13,15)(14,18)(16,23)(17,22)(19,21)(20,26)(24,25)(27,28);(1,2,4,3)(5,7,9,8)(6,11,10,12);(1,3,4,2)(15,22)(17,25)(18,26)(20,28)(21,27)(23,24);(5,8,9,7)(6,11,10,12)(14,19)(15,23)(20,28)(22,24)

Polycyclic presentation:

2^9|a^2=f,c^2=fgi,d^2=g,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,h^d=hi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307411, #72307564, #72307879
maximal normal subgroup of	#72307411, #72307564, #72307879

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7471819);