#196853

order	512
nilpotency class	3
isoclinism	isoclinic class of #162406
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#35165	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1638 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,c^2=hi,d^2=h,e^2=hi,g^2=hi,b^a=be,c^a=cf,e^a=eh,e^b=ei,g^b=gi,d^c=dh

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 104049);