#495308

order	512
nilpotency class	4
isoclinism	isoclinic class of #33605
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₁₆	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

636 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=fhi,d^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^c=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318464, #72318470, #72318484, …
maximal normal subgroup of	#72318464, #72318470, #72318484, …
maximal quotient of	#72322452

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402504);