#2280908

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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

2904 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fhi,c^2=fi,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72331979, #72332008, #72332044
maximal normal subgroup of	#72331979, #72332008, #72332044
maximal quotient of	#72332983

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2188104);