#414957

order	512
nilpotency class	3
isoclinism	isoclinic class of #393904
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1468 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=efh,d^2=h,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=di,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239014, #72239144, #72239151
maximal normal subgroup of	#72239014, #72239144, #72239151

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 322153);