#1007457

order	512
nilpotency class	2
isoclinism	isoclinic class of #988785
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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

3190 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=f,c^2=h,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,e^c=ef

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72332178, #72332181, #72332277
maximal normal subgroup of	#72332178, #72332181, #72332277
maximal quotient of	#72333107

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 914653);