#357802

order	512
nilpotency class	3
isoclinism	isoclinic class of #23387
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴×D₄	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3446 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308430, #72308457, #72308587, …
maximal normal subgroup of	#72308430, #72308457, #72308587, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 264998);