#355013

order	512
nilpotency class	3
isoclinism	isoclinic class of #3055
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1214 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=ef,d^2=ehi,f^2=i,b^a=be,c^a=cf,f^a=fi,c^b=ci,d^c=di

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240741, #72240744, #72240759, …
maximal normal subgroup of	#72240741, #72240744, #72240759, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 262209);