#3867472

order	512
nilpotency class	2
isoclinism	isoclinic class of #3858029
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

3010 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)(5,6,10,12)(7,9,8,11)(13,14)(15,19)(16,17)(18,20)(21,22,24,26)(23,27,28,25);(5,7,10,8)(6,11,12,9)(14,17)(18,19)(21,22,24,26)(23,25,28,27);(5,7,10,8)(6,11,12,9)(13,14,16,17)(15,18,20,19)(21,23)(22,25)(24,28)(26,27);(1,2)(3,4)(5,8,10,7)(6,9,12,11)(13,15)(14,18)(16,20)(17,19)(21,22)(23,25)(24,26)(27,28);(5,9)(6,8)(7,12)(10,11)(13,14)(15,19)(16,17)(18,20)(21,23)(22,25)(24,28)(26,27)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=f,d^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,e^b=ef,e^c=ef,e^d=ef

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72332349, #72332358, #72332377
maximal normal subgroup of	#72332349, #72332358, #72332377
maximal quotient of	#72332983

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3774668);