#367069

order	512
nilpotency class	3
isoclinism	isoclinic class of #25443
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×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×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1500 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)(5,6,7,8)(9,10,14,19)(11,20,22,17)(12,21,23,24)(13,16,15,18);(1,2)(3,4)(10,16,19,18)(11,21,22,24)(12,23)(17,20);(2,4)(6,8)(9,11)(10,17)(12,16)(13,21)(14,22)(15,24)(18,23)(19,20);(9,12,15,20,14,23,13,17)(10,11,16,24,19,22,18,21)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72216140
maximal normal subgroup of	#72216140
maximal quotient of	#72216101

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 274265);