#355920

order	512
nilpotency class	3
isoclinism	isoclinic class of #17392
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁶	C₂×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁵×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3046 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=ce,g^c=gi

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, 263116);