#2318453

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

2684 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)(7,12)(8,11)(9,10)(14,16)(15,19)(22,25)(23,27);(5,7)(6,8)(9,12)(10,11)(15,19)(18,20)(21,22)(23,26)(24,25)(27,28);(6,9)(7,11)(13,14,17,16)(15,18,19,20)(21,23,24,27)(22,26,25,28);(1,2,3,4)(5,8,10,12)(6,11,9,7)(13,15,17,19)(14,18,16,20)(21,22)(23,26)(24,25)(27,28);(5,9,10,6)(7,12,11,8)(13,16,17,14)(15,20,19,18)(21,23)(22,26)(24,27)(25,28)

Polycyclic presentation:

2^9|c^2=gh,d^2=fi,e^2=fgh,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2225649);