#245273

order	512
nilpotency class	3
isoclinism	isoclinic class of #245246
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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

1620 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)(3,6)(4,11)(5,8)(7,12)(9,14)(10,13)(15,16)(17,18)(19,22)(20,23)(21,24)(25,29)(26,28)(27,30)(31,32);(6,13)(14,16)(17,19)(18,22)(20,25)(21,26)(23,29)(24,28)(27,31)(30,32);(1,3)(2,6)(4,9)(5,10)(7,14)(8,13)(11,16)(12,15)(22,28)(29,32);(1,4,5,12)(2,7,8,11)(3,9,10,15)(6,14,13,16)(17,20,21,27)(18,23,24,30)(19,25,26,31)(22,29,28,32)

Polycyclic presentation:

2^9|d^2=hi,b^a=be,c^a=cf,d^a=deh,g^a=ghi,c^b=cg,f^b=fi,e^c=eh,f^d=fh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240624, #72240625, #72240628, …
maximal normal subgroup of	#72240624, #72240625, #72240628, …
maximal quotient of	#72251437, #72251440, #72251464, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 152469);