#4308016

order	512
nilpotency class	2
isoclinism	isoclinic class of #4307937
automorphism	Aut(G) order 2097152
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	C₂×#2631	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3192 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4215212);