#809140

order	512
nilpotency class	2
isoclinism	isoclinic class of #772778
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2184 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=fg,c^2=gi,d^2=fg,e^2=fgh,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,e^d=ef

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72304042, #72304082, #72304113
maximal normal subgroup of	#72304042, #72304082, #72304113

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 716336);