#365112

order	512
nilpotency class	3
isoclinism	isoclinic class of #24384
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2026 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fh,f^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,f^c=fi,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72309778, D₄×#1938, #72310137
maximal normal subgroup of	#72309778, D₄×#1938, #72310137

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 272308);