#2929367

order	512
nilpotency class	2
isoclinism	isoclinic class of #2805613
automorphism	Aut(G) order 1048576
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

3136 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72328784, #72328805, #72328851
maximal normal subgroup of	#72328784, #72328805, #72328851
maximal quotient of	#72330485

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2836563);