#94075

order	512
nilpotency class	5
isoclinism	isoclinic class of #7339
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1316
commutator	C₂²×C₄	C₄×C₈
Frattini, cosocle	#2280	C₂²
socle	C₂	#7187
prime radical	C₂⁵	C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ #2280 ◁ G
LECS	G ▷ #2280 ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁

408 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 64 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 2 generators on 32 points:

(1,2,5,10)(3,12,14,29,15,23,30,32)(4,16,18,27)(6,9,19,25)(7,11,20,28)(8,21,22,17,24,31,26,13);(1,3,6,13,7,15,4,17)(2,8,9,23,11,24,16,12)(5,14,19,21,20,30,18,31)(10,26,25,29,28,22,27,32)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=i,d^2=g,e^2=i,b^a=bc,c^a=cf,f^a=fh,h^a=hi,c^b=ci,d^b=dfi,g^b=gi,d^c=dh,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1271);