#248568

order	512
nilpotency class	3
isoclinism	isoclinic class of #248551
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#546	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1353 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=hi,e^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=gi,c^b=cg,d^b=dh,e^b=eh,f^b=fi,e^d=eh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72268416, #72268419, #72268429, …
maximal normal subgroup of	#72268416, #72268419, #72268429, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 155764);