#154820

order	512
nilpotency class	2
isoclinism	isoclinic class of #15257
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁵	C₂⁴
commutator	C₂⁴	C₂³×C₄
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

3778 conjugacy classes of subgroups.

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

Polycyclic presentation:

2^9|a^2=i,b^2=fh,c^2=ef,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72328484, #72328494, #72328505, …
maximal normal subgroup of	#72328484, #72328494, #72328505, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 62016);