#189661

order	512
nilpotency class	3
isoclinism	isoclinic class of #184831
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

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 ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1036 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,d^2=hi,g^2=i,b^a=be,c^a=cf,f^a=fh,d^b=dh,d^c=di,g^c=gh

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, 96857);