#205469

order	512
nilpotency class	3
isoclinism	isoclinic class of #19456
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#3207	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1160 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=eghi,c^2=f,d^2=hi,e^2=hi,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=dh,e^b=ei,g^b=gi,d^c=dhi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 112665);