#94078

order	512
nilpotency class	5
isoclinism	isoclinic class of #7348
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1317
commutator	C₂²×C₄	C₄×C₈
Frattini, cosocle	C₂×#407	C₂²
prime radical	C₂⁵	M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#407 ◁ G
LECS	G ▷ C₂×#407 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

412 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 66 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) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 2 generators on 36 points:

(5,6,9,14,10,13,23,29,11,15,22,32,8,20,21,31)(7,16,18,33,17,12,25,26,19,28,34,36,24,27,35,30);(1,2,3,4)(5,7)(6,12,15,27)(8,17)(9,18)(10,24)(11,19)(13,16,20,28)(14,30,32,26)(21,25)(22,34)(23,35)(29,36,31,33)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=i,d^2=g,g^2=i,b^a=bc,c^a=cf,e^a=ei,f^a=fh,h^a=hi,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, 1274);