#461252

order	512
nilpotency class	3
isoclinism	isoclinic class of #461251
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2644
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

848 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)(7,17,18,26)(8,22)(9,15)(10,13)(11,16)(12,20,28,21)(14,25)(19,33,32,31)(23,29)(24,30)(27,34,36,35);(5,7,10,21,11,18,9,20)(6,12,15,17,16,28,13,26)(8,19,24,34,25,32,23,35)(14,27,30,31,22,36,29,33);(6,13)(7,18)(8,23)(9,10)(12,26)(15,16)(17,28)(19,34)(24,25)(29,30)(31,33)(32,35);(2,4)(5,8,10,24,11,25,9,23)(6,14,13,29,16,22,15,30)(7,19,21,34,18,32,20,35)(12,27,26,33,28,36,17,31)

Polycyclic presentation:

2^9|a^2=efg,b^2=f,d^2=ei,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,g^b=gi,d^c=de,e^c=ei,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 368448);