#195971

order	512
nilpotency class	3
isoclinism	isoclinic class of #195963
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1219 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2)(3,4)(5,6,9,12,10,13,8,17)(7,11,15,18,16,19,14,20)(21,22,26,30,25,29,34,35)(23,27,28,36,31,24,32,33);(5,7,9,15,10,16,8,14)(6,11,12,18,13,19,17,20)(21,23,26,32,25,31,34,28)(22,27,35,36,29,24,30,33);(2,4)(21,24,25,27)(22,28,29,32)(23,30,31,35)(26,33,34,36);(5,8,10,9)(6,12,13,17)(7,14,16,15)(11,18,19,20)(22,29)(24,27)(30,35)(33,36)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=h,d^2=hi,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,e^b=eh,g^b=gh,d^c=dh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 103167);