#272857

order	512
nilpotency class	3
isoclinism	isoclinic class of #272795
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1266 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=h,c^2=hi,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,e^b=ei,f^b=fhi,e^c=ehi,e^d=ei,f^d=fhi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 180053);