#202425

order	512
nilpotency class	3
isoclinism	isoclinic class of #24378
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

824 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,f^a=fh,d^c=di,f^c=fh

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 109621);