#33326

order	256
nilpotency class	2
isoclinism	isoclinic class of H(5,𝔽₂)
automorphism	Aut(G) order 24576
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂×C₈	C₂⁴
commutator	C₂	C₂²×C₄×C₈
socle	C₂²	C₂³×C₈
prime radical	C₂×#41	C₂×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

409 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^8|a^2=e,b^2=f,e^2=g,g^2=h,b^a=bh,d^c=dh

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

Frattini subgroup of	#72286349, #72286399, #72286437, …
Fitting subgroup of	#33326⋊C₃
maximal subgroup of	#128258, #128270, #134661, …
maximal normal subgroup of	#128258, #128270, #134661, …
maximal quotient of	#479899, #479903, #479917, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 26314);