#2044043

order	512
nilpotency class	2
isoclinism	isoclinic class of #2031997
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂³×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3187 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=fgh,d^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72326790, #72326826, #72326872
maximal normal subgroup of	#72326790, #72326826, #72326872
maximal quotient of	#72328339

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1951239);