#6396153

order	512
nilpotency class	3
isoclinism	isoclinic class of #62283
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3198	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2130 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gh,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ci,d^b=df,d^c=di,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6303349);