#2761359

order	512
nilpotency class	2
isoclinism	isoclinic class of #2760365
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#2410	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3795 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72327160, #72327223, #72327264
maximal normal subgroup of	#72327160, #72327223, #72327264
maximal quotient of	#72328442

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2668555);