#449468

order	512
nilpotency class	3
isoclinism	isoclinic class of #417442
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

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

3476 conjugacy classes of subgroups; only the 320 normal subgroups are shown.

normal subgroups only

This group has 3476 conjugacy classes of subgroups — too many to draw. Only its 320 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) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,e^b=ei,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227967, #72227982
maximal normal subgroup of	#72227967, #72227982

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 356664);