#362081

order	512
nilpotency class	3
isoclinism	isoclinic class of #24805
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#389
prime radical	C₂⁵	C₂×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1270 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=f,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,d^b=di,g^c=gi

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, 269277);