#6392695

order	512
nilpotency class	3
isoclinism	isoclinic class of #61643
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×#525	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2088 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)(15,22)(16,17)(19,26)(20,24)(23,25)(27,28);(5,7)(6,9)(8,12)(10,11)(13,14,18,21)(15,19,23,28)(16,24,17,20)(22,26,25,27);(1,3,4,2)(13,15,18,23)(14,19,21,28)(16,22,17,25)(20,27,24,26);(7,12)(9,11)(13,16,18,17)(14,20,21,24)(15,22,23,25)(19,27,28,26);(6,10)(9,11)(14,21)(19,28)(20,24)(26,27)

Polycyclic presentation:

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

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, 6299891);