#515266

order	512
nilpotency class	2
isoclinism	isoclinic class of #2613
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂⁴
commutator	C₂³	C₂⁴×C₄
Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2722 conjugacy classes of subgroups; only the 1090 normal subgroups are shown.

normal subgroups only

This group has 2722 conjugacy classes of subgroups — too many to draw. Only its 1090 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) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=i,e^2=f,b^a=bf,c^a=cg,c^b=ch,d^c=df

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72293232, #72293247, #72293260
maximal normal subgroup of	#72293232, #72293247, #72293260

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 422462);