#212874

order	512
nilpotency class	3
isoclinism	isoclinic class of #24384
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂²×D₄	C₂²×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

750 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=fh,d^2=g,e^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,c^b=chi,d^b=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 120070);