#183251

order	512
nilpotency class	3
isoclinism	isoclinic class of #17603
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1010 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 90447);