#182715

order	512
nilpotency class	3
isoclinism	isoclinic class of #17577
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 ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1110 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)(7,15)(8,13)(9,12)(10,14)(11,18)(16,20)(17,19)(21,22,24,27)(23,29,30,26)(25,33,32,28)(31,34,36,35);(5,7)(6,11,14,19)(8,17)(9,18)(10,16)(12,20,13,15)(21,23,25,31,24,30,32,36)(22,26,28,34,27,29,33,35);(1,3,4,2)(7,16)(11,19)(15,20)(17,18);(5,8,10,9)(6,12,14,13)(7,17,16,18)(11,20,19,15)

Polycyclic presentation:

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

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, 89911);