#386792

order	512
nilpotency class	3
isoclinism	isoclinic class of #30484
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1404 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,c^2=fg,d^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,e^b=ei,g^b=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72215020, #72216143, #72217009
maximal normal subgroup of	#72215020, #72216143, #72217009
maximal quotient of	#72214933, #72214947

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 293988);