#7522985

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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	#25081	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1920 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,d^2=fi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,h^d=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257549, #72257588, #72263873
maximal normal subgroup of	#72257549, #72257588, #72263873

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7430181);